Anomalies and chiral symmetry in QCD

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by  Michael Creutz

Anomalies and chiral symmetry in QCD
Michael Creutz
Physics Department, Brookhaven National Laboratory
Upton, NY 11973, USA
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Abstract
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I review some aspects of the interplay between anomalies and chiral symme-
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try. The quantum anomaly that breaks the U(1) axial symmetry of massless

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QCD leaves behind a flavor-singlet discrete chiral invariance. When the mass
h
p is turned on this residual symmetry has a close connection with the strong
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p CP violating parameter theta. One result is that a first order transition is
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usually expected when the strong CP violating angle passes through pi. This
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[ symmetry can be understood either in terms of effective chiral Lagrangians

or in terms of the underlying quark fields.
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Key words: Chiral symmetry, quark masses, anomalies
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5 PACS: 11.30.Rd, 12.39.Fe, 11.15.Ha, 11.10.Gh
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1. Introduction
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The classical Lagrangian for QCD couples left and right handed quark
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: fields only through mass terms. Thus naively the massless theory will have
v
i independent conserved currents associated with each handedness. For N
X f
massless flavors, this would be an independent U(N ) symmetry associated
r f
a witheachchirality, givingafullsymmetry thatisoftenwrittenintermsofax-
ial and vector fields as U(N ) ×U(N ) . As is well known, this full symme-
f V f A
try does not survive quantization, being broken to a SU(N ) ×SU(N ) ×
f V f A
U(1) , where the U(1) represents the symmetry of baryon number con-
B B
servation. The only surviving axial symmetries of the quantum theory are
non-singlet under flavor symmetry.
ThisbreakingoftheclassicalU(1)axialsymmetryistiedtothepossibility
of introducing into massive QCD a CP violating parameter, usually called
Θ. For an extensive recent review of this quantity, see Ref. [1]. While such
a term is allowed from fundamental principles, experimentally it appears to
Preprint submitted to Elsevier January 27, 2009
be extremely small. This raises an unresolved puzzle for attempts to unify
the strong interactions with the weak. Since the weak interactions do violate
CP, why is there no residue of this remaining in the strong sector below the
unification scale?
One goal here is to provide a qualitative picture of the Θ parameter in
meson physics. I will concentrate on symmetry alone and will not attempt to
rely on any specific form for an effective Lagrangian. I build on a connection
between Θ and a flavor-singlet Z symmetry that survives the anomaly.
Nf
This symmetry predicts that, if the lightest quarks are massive and degener-
ate, then a first order transition is expected when Θ passes through π. This
transition is quite generic, and only can be avoided under limited conditions
with one quark considerably lighter than the others. It will also become clear
why the sign of the quark mass is relevant for an odd number of flavors, an
effect unseen in naive perturbation theory.
This picture has evolved over many years. The possibility of the sponta-
neous CP violation occurring at Θ = π is tied to what is known as Dashen’s
phenomenon [2], first noted even before the days of QCD. In the mid 1970’s,
’t Hooft [3] elucidated the underlying connection between the chiral anomaly
and the topology of gauge fields. Later Witten [4] used large gauge group
ideas to discuss the behavior at Θ = π in terms of effective Lagrangians.
Ref. [5] lists a few of the early studies of the effects of Θ on effective La-
grangians via a mixing between quark and gluonic operators. The topic
continues to appear in various contexts; for example, Ref. [6] contains a dif-
ferent approach to understanding the transition at Θ = π via the framework
of the two-flavor Nambu Jona-Lasinio model.
I became interested in the issues while trying to understand difficulties
with formulating chiral symmetry on the lattice. Much of the picture pre-
sented here is implicit in my 1995 paper on quark masses [7]. Since then
the topic has become highly controversial, with the realization of ambigui-
ties precluding a vanishing up quark mass solving the strong CP problem [8]
and the appearance of an inconsistency with one of the popular algorithms
in lattice gauge theory [9]. Despite the controversies, both are direct conse-
quences of the interplay of the anomaly and chiral symmetry discussed here.
The fact that these issues remain so disputed is what has driven me to write
this overview.
Section 2 reviews the conventional picture of spontaneous chiral symme-
try breaking wherein the light pseudoscalars are identified as approximate
Goldstone bosons in an effective Lagrangian. Here I introduce a mass con-
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tribution, coming from the anomaly, for for the flavor singlet pseudoscalar
meson. Section 3 reviews how the anomaly arises in terms of the underlying
quark fields and gauge fields with non-trivial topology. Here the fact that the
theory requires a regulator that breaks chiral symmetry is crucial. Section 4
returns to the effective field picture and exposes the flavor singlet Z sym-
Nf
metry. InSection5 Iaddina smallquark mass tobreak thechiralsymmetry.
Depending on the mass and N , the effective potential can display multiple
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meta-stable minima. Doing an anomalous chiral rotation on the mass term
brings in the parameter Θ. The first order transition at Θ = π corresponds
to a jump of the physical vacuum between two distinct degenerate minima.
At this point the theory spontaneously breaks CP invariance. Section 6 dis-
cusses increasing the mass of one quark species to allow an interpolation
between various N . Section 7 draws concluding remarks.
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I start with a few reasonably uncontroversial assumptions. First QCD
with N light quarks should exist as a field theory and exhibit confinement in
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the usual way. I assume the validity of the standard picture of chiral symme-
try breaking involving a quark condensate hψψi =6 0. The conventional chiral
perturbation theory based on expanding in masses and momenta around the
chiral limit should make sense. I assume the usual result that the anomaly
generates a mass for the η′ particle with this mass surviving in the chiral
limit. And I consider N small enough to avoid any potential conformal
f
phase of QCD [10].
Throughout I use the language of continuum field theory. I do have
in mind that some non-perturbative regulator has been imposed to define
various products of fields, such as the condensing combination σ = ψψ.
For a momentum space cutoff, I assume that it is much larger than Λ .
QCD
Correspondingly, for a lattice cutoff, then I imagine that the lattice spacing
is much smaller than 1/Λ . Thus I ignore any lattice artifacts that are
QCD
expected to vanish in the continuum limit.
2. Effective potentials
I begin by considering an effective potential V as a function of the various
mesonfields intheproblem. Intuitively, V represents theenergyofthelowest
state for a given field expectation. More formally, this can be defined in the
standard way via a Legendre transformation. I will ignore the well known
result that effective potentials must be convex functions of their arguments.
This is easily understood in terms of a Maxwell construction involving the
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phase separation that will occur if one asks for a field expectation in what
would otherwise be a concave region. Ignoring this effect allows me to use
the normal language of spontaneous symmetry breaking corresponding to
having an effective potential with more than one minimum. When the un-
derlying theory possesses some symmetry but the individual minima do not,
spontaneous breaking comes about when the vacuum selects one minimum
arbitrarily.
Iframethe discussion interms of compositescalar andpseudoscalar fields
σ ∼ ψψ
π ∼ iψλ γ ψ
α α 5 (1)
η′ ∼ iψγ ψ
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δ ∼ ψλ ψ
α α
Heretheλ arethegeneralizationoftheusualGell-MannmatricestoSU(N ).
α f
The δ field is listed here for completeness, but will not play a role in the fol-
lowing discussion. As mentioned earlier, I assume some sort of regulator,
perhaps a lattice, is in place to define these products of fields at the same
point.
Initially consider degenerate quarks with a small common mass m. I also
beginby restricting N tobe even, returning later to thesubtleties arising for
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an odd number of flavors. And, as mentioned earlier, I keep N small enough
f
to maintain asymptotic freedom as well as to avoid any possible conformal
phases.
The conventional picture of spontaneous chiral symmetry breaking at
m = 0 begins with the vacuum acquiring a quark condensate with hψψi =
hσi = v 6= 0. In terms of the effective potential, V(σ) should acquire a
double well structure, as sketched in Fig. 1. The symmetry under σ ↔ −σ is
associated with the invariance of the action under a flavored chiral rotation.
For example, with two flavors the change of variables
ψ → eiπτ3γ5/2ψ
(2)
ψ → ψeiπτ3γ5/2
leaves the massless action invariant but changes the sign of σ. Here τ is the
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conventional Pauli matrix corresponding to the third component of isospin.
Extending the effective potential to a function of the non-singlet pseu-
doscalar fields gives the standard picture of Goldstone bosons. These are
massless when the quark mass vanishes, corresponding to N2−1 “flat” direc-
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tions for the potential. It is useful to introduce the SU(N ) valued effective
f
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V
σ
Figure 1: Spontaneous chiral symmetry breaking is represented by a double well effective
potential with the vacuum settling into one of two possible minima. In this minimum
chiral symmetry is broken by the selection of a specific value for the quark condensate.
V
π
σ
Figure2: The flavornon-singletpseudoscalarmesonsareGoldstonebosonscorresponding
to flat directions in the effective potential.
field
Σ = eiλαπα/Fπ ∼ ψ ψ (3)
L R
where the matrices λ generalize of the Gell-mann matrices to SU(N ), flavor
f
indices that make this a matrix quantity are suppressed, and the pion decay
constant F is inserted as a convenient normalization but will play no role in
π
the qualitative discussion here. The left and right Fermi fields are defined as
usual by ψ = 1∓γ5ψ. In terms of Σ the chiral invariance of the potential
L,R 2
takes the simple form
†
V(Σ) = V(g Σg ) (4)
L R
where g and g are arbitrary elements of SU(N ). One flat direction is
L R f
sketched in Fig. 2.
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V
π
σ
Figure 3: A small quark mass term tilts the effective potential, selecting one direction for
the true vacuum and giving the Goldstone bosons a mass.
The introduction of a small mass for the quarks effectively tilts the poten-
tial V(σ) → V(σ)−mσ. This selects one of the minima as the true vacuum,
driving the above matrix Σ → I. The tilting of the potential breaks the
global symmetry and gives the Goldstone bosons a small mass proportional
to the square root of the quark mass, as sketched in Fig. 3
This picture is, of course, completely standard. It is also common lore
that the anomaly prevents the η′ from being a Goldstone boson and leaves
it with a mass of order Λ even in the massless quark limit. The effective
QCD
potential V must not be symmetric under the following anomalous rotation
by an angle φ
σ → σcos(φ)+η′sin(φ)
(5)
η′ → −σsin(φ)+η′cos(φ).
If we consider the effective potential as a function of the fields σ and η′,
it should have a minimum at σ ∼ v and η′ ∼ 0. Expanding about that point
we expect a qualitative form
V(σ,η′) ∼ m2(σ −v)2 +m2 η′2 +O((σ−v)3,η′4) (6)
σ η′
where both mσ and mη′ remain of order ΛQCD, even in the chiral limit. And,
at least with an even number of flavors as considered here, there should be
a second minimum with σ ∼ −v. Expanding about this point gives the
V(σ,η′) ∼ m2(σ +v)2 +m2 η′2 +O((σ+v)3,η′4). (7)
σ η′
At this point one can ask whether we know anything else about the effec-
tive potential in this (σ,η′) plane. In section 4 I show that indeed we do, and
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the potential has a total of N equivalent minima in the chiral limit. But
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first I digress to review how the above minima arise in quark language.
3. Quark fields
The classical QCD Lagrangian has a symmetry under a rotation of the
underlying quark fields
ψ → eiφγ5/2ψ
(8)
ψ → ψeiφγ5/2
This corresponds directly to the transformation of the composite fields given
in Eq. 5. This symmetry is “anomalous” and thus any regulator must break
it with a remnant surviving in the continuum limit. The specifics of how this
works depend on the details of the regulator, but a simple understanding
[11] comes from considering the fermionic measure in the path integral. If
we make the above rotation on the field ψ, the measure changes by the
determinant of the rotation matrix
dψ → |e−iφγ5/2|dψ = e−iφTrγ5/2dψ. (9)
Here the subtlety of the regulator comes in. Naively γ is a simple four
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by four traceless matrix. If it is indeed traceless, then the measure would
be invariant. However in the regulated theory this is not the case. This is
intimately tiedwith theindex theorem fortheDiracoperatorintopologically
non-trivial gauge fields.
A typical Diracaction takes the form ψ(D+m)ψ with D a function of the
gauge fields. In the naive continuum theory D is anti-Hermitian, D† = −D,
and anti-commutes with γ , i.e. [D,γ ] = 0. What complicates the issue
5 5 +
with fermions is the well known index theorem: if a background gauge field
has winding ν, then there are known to be at least ν exact zero eigenvalues
of D. Furthermore, on the space spanned by the corresponding eigenvectors,
γ can be simultaneously diagonalized with D. The net winding number
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equals the number of positive eigenvalues of γ minus the number of negative
5
eigenvalues. This theorem is well known and well reviewed elsewhere [12].
Here I only will use the fact that in this subspace the trace of γ does not
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vanish, but equals ν.
What about the higher eigenvalues of D? Because [D,γ ] = 0, these
5 +
appear in opposite sign pairs; i.e. if D|ψi = λ|ψi then Dγ |ψi = −λγ |ψi.
5 5
For an anti-Hermitean D, these modes are orthogonal with hψ|γ ψi = 0.
5
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As a consequence, γ is traceless on the subspace spanned by each pair of
5
eigenvectors.
So what happened to the opposite chirality states to the zero modes? In
a regulated theory they are in some sense “above the cutoff.” In a simple
continuum discussion they have been “lost at infinity.” With a lattice regu-
lator there is no “infinity”; so, something more subtle must happen. With
the overlap[13] or Wilson[14] fermions, one gives up the anti-Hermiticity of
D. Most eigenvalues still occur in conjugate pairs and do not contribute to
the trace of γ . However, in addition to the small real eigenvalues repre-
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senting the zero modes, there are additional modes where the eigenvalues are
large and real. With Wilson fermions these appear as massive doubler states.
With the overlap, the eigenvalues are constrained to lie on a circle. In this
case, for every exact zero mode there is another mode of opposite chirality
lying on the opposite side of the circle. These modes are effectively massive
and break chiral symmetry.
So with the regulator in place, the trace of γ does not vanish on gauge
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configurations of non-trivial topology. The change of variables indicated in
Eq. 9 introduces into the path integral a modification of the weighting by a
factor
e−iφTrγ5 = e−iφNfν (10)
here I consider applying the rotation to all flavors equally, thus the factor of
N in the exponent. The conclusion is that gauge configurations that have
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non-trivial topology receive a complex weight after the anomalous rotation.
This changes the underlying physics and gives an inequivalent theory. Al-
though not the topic of discussion here, note that this factor introduces a
sign problem if one wishes to study this physics via Monte Carlo simulations.
Here I have treated all N flavors equivalently; this corresponds to dividing
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the conventionally defined CP violation angle, to be discussed later, equally
among the flavors, i.e. effectively φ = Θ/N .
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The necessary involvement of both small and large eigenvalues warns of
the implicit danger in attempts to separate infrared from ultraviolet effects.
When the anomaly is concerned, going to short distances is not sufficient for
ignoring non-perturbative effects related to topology.
4. A discrete chiral symmetry
I now return to the effective Lagrangian language of before. For the
massless theory, the symmetry under σ ↔ −σ indicates that we expect at
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η
?
σ
V V
1 0
?
′ ′
Figure 4: We have two minima in the σ,η plane located at σ = ±v and η = 0. Can we
find any other minima?
least two minima for the effective potential considered in the σ,η′ plane.
These are located as sketched in Fig. 4. Do we know anything about the
potentialelsewhere inthisplane? Theanswer isyes, indeedthereareactually
N equivalent minima.
f
As noted before, due to the anomaly the singlet rotation
ψ → eiφψ (11)
L L
is not a valid symmetry of the theory for generic values of the angle φ. On
the other hand, flavored chiral symmetries should survive, and in particular
ψ → g ψ = eiφαλαψ (12)
L L L L
should be a valid symmetry for any set of angles φ . The point of this
α
section is that, for special special discrete values of the angles, the rotations
in Eq. 11 and Eq. 12 can coincide. At such values the singlet rotation is a
valid symmetry. In particular, note that
g = e2πiφ/Nf ∈ Z ⊂ SU(N ). (13)
Nf f
Thus a valid discrete symmetry involving only σ and η′ is
σ → σcos(2π/N )+η′sin(2π/N )
f f (14)
η′ → −σsin(2π/N )+η′cos(2π/N ).
f f
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η
V
1
σ
V V
2 0
V
3
′
Figure 5: For four flavors we have four equivalent minima in the σ,η plane. This gener-
alizes to N minima with N flavors.
f f
The potential V(σ,η′) has a Z symmetry manifested in N equivalent min-
Nf f
ima in the (σ,η′) plane. For four flavors this structure is sketched in Fig. 5.
This discrete flavor singlet symmetry arises from the trivial fact that Z
N
is a subgroup of both SU(N) and U(1). At the quark level the symme-
try is easily understood since the ’t Hooft vertex, responsible for the chiral
anomaly, receives one factor from every flavor. With N flavors, these mul-
F
tiply together making
ψ → e2πi/Nfψ (15)
L L
a valid symmetry even though rotations by smaller angles are not.
The role of the Z center of SU(N) is illustrated graphically in Fig. 6,
N
taken from Ref. [7]. Here I plot the real and the imaginary parts of the traces
of 10,000 SU(3) matrices drawn randomly with the invariant group measure.
The region of support only touches the U(1) circle at the elements of the
center. All elements lie on or within the curve mapped out by elements of
form exp(iφλ ). Fig. 7 is a similar plot for the group SU(4).
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5. Massive quarks
As discussed earlier and illustrated in Fig.5, a quark mass term −mψψ ∼
−mσ isrepresentedbya“tilting”oftheeffectivepotential. Thisselectsoneof
the minima in the σ,η′ plane as the true vacuum. For masses small compared
to the scale of QCD, the other minima will persist, although due to the flat
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